Weighted A-Statistical Convergence for Sequences of Positive Linear Operators

We introduce the notion of weighted A-statistical convergence of a sequence, where A represents the nonnegative regular matrix. We also prove the Korovkin approximation theorem by using the notion of weighted A-statistical convergence. Further, we give a rate of weighted A-statistical convergence and apply the classical Bernstein polynomial to construct an illustrative example in support of our result.


Background, Notations, and Preliminaries
We begin this paper by recalling the definition of natural (or asymptotic) density as follows. Suppose that ⊆ N := {1, 2, . . .} and = { ≤ : ∈ }. Then is called the natural density of provided that the limit exists, where | ⋅ | represents the number of elements in the enclosed set. The term "statistical convergence" was first presented by Fast [1] which is a generalization of the concept of ordinary convergence. Actually, a root of the notion of statistical convergence can be detected by Zygmund [2] (also, see [3]), where he used the term "almost convergence" which turned out to be equivalent to the concept of statistical convergence. The notion of Fast was further investigated by Schoenberg [4], Salát [5], Fridy [6], and Conner [7].
The following notion is due to Fast [1]. A sequence = ( ) is said to be statistically convergent to if ( ) = 0 for every > 0, where Equivalently, In symbol, we will write -lim = . We remark that every convergent sequence is statistically convergent but not conversely.
Let and be two sequence spaces and let = ( , ) be an infinite matrix. If for each = ( ) in the series converges for each ∈ N and the sequence = ( ) belongs to , then we say that matrix maps into . By the symbol ( , ) we denote the set of all matrices which map into .
A matrix (or a matrix map ) is called regular if ∈ ( , ), where the symbol denotes the spaces of all convergent sequences and for all ∈ . The well-known Silverman-Toeplitz theorem (see [8]) asserts that = ( , ) is regular if and only if The Scientific World Journal Kolk [9] extended the definition of statistical convergence with the help of nonnegative regular matrix = ( , ) calling it -statistical convergence. The definition of -statistical convergence is given by Kolk as follows. For any nonnegative regular matrix , we say that a sequence is -statistically convergent to provided that for every > 0 we have In 2009, the concept of weighted statistical convergence was defined and studied by Karakaya and Chishti [10] and further modified by Mursaleen et al. [11] in 2012. In 2013, Belen and Mohiuddine [12] presented a generalization of this notion through de la Vallée-Poussin mean. Quite recently, Esi [13] defined and studied the notion statistical summability through de la Vallée-Poussin mean in probabilistic normed spaces. Let = ( ) be a sequence of nonnegative numbers such that 0 > 0 and We say that the sequence = ( ) is ( , )-summable to if lim → ∞ = .
The lower and upper weighted densities of ⊆ N are defined by respectively. We say that has weighted density, denoted by ( ), if the limits of both of the above densities exist and are equal; that is, one writes The sequence = ( ) is said to be weighted statistically convergent (or -V ) to if, for every > 0, the set { ∈ N : | − | ≥ } has weighted density zero; that is, In this case we write = -lim .
On the other hand, let us recall that C[ , ] is the space of all functions continuous on [ , ]. We know that C[ , ] is a Banach space with norm Suppose that is a linear operator from It is clear that if ≥ 0 implies ≥ 0, then the linear operator is positive on C[ , ]. We denote the value of at a point ∈ [ , ] by ( ; ). The classical Korovkin approximation theorem states the following [14].
Many mathematicians extended the Korovkin-type approximation theorems by using various test functions in several setups, including Banach spaces, abstract Banach lattices, function spaces, and Banach algebras. Firstly, Gadjiev and Orhan [15] established classical Korovkin theorem through statistical convergence and display an interesting example in support of our result. Recently, Korovkin-type theorems have been obtained by Mohiuddine [16] for almost convergence. Korovkin-type theorems were also obtained in [17] for -statistical convergence. The authors of [18] established these types of approximation theorem in weighted spaces, where 1 ≤ < ∞, through -summability which is stronger than ordinary convergence. For these types of approximation theorems and related concepts, one can be referred to [19][20][21][22][23][24][25][26][27] and references therein.

Korovkin-Type Theorems by Weighted
-Statistical Convergence Kolk [9] introduced the notion of -statistical convergence by considering nonnegative regular matrix instead of Cesáro matrix in the definition of statistical convergence due to Fast. Inspired from the work of Kolk, we introduce the notion of weighted -statistical convergence of a sequence and then we establish some Korovkin-type theorems by using this notion.
The Scientific World Journal In symbol, we will write -lim = .
Remark 4. One has the following.
(i) If we take = , where denotes the identity matrix, then weighted -statistical convergence of a sequence is reduced to ordinary convergence.
(ii) If we take = ( , 1), where ( , 1) denotes the Cesáro matrix of order one, then weighted -statistical convergence of a sequence reduces to weighted statistical convergence.
(iii) If we take = ( , 1) and = 1 for all , then weighted -statistical convergence of a sequence reduces to statistical convergence.
Note that convergent sequence implies weightedstatistically convergent to the same value but the converse is not true in general. For example, take = ( , 1) and = 1 for all and define a sequence = ( ) by where ∈ N. Then this sequence is statistically convergent to 0 but not convergent; in this case, weighted -statistical convergence of a sequence coincides with statistical convergence.
This yields that for all ∈ C[ , ].
We also obtain the following Korovkin-type theorem for weighted statistical convergence by writing Cesáro matrix ( , 1) instead of nonnegative regular matrix in Theorem 5.
Proof. Following the proof of Theorem 5, one obtains and so Equations (42)-(44) give that The Scientific World Journal 5 Remark 7. If we replace nonnegative regular matrix by Cesáro matrix and choose = 1 for all , in Theorem 5, then we obtain Theorem 1 due to Gadjiev and Orhan [15].

Remark 8. By Theorem 2 of [10], we have that if a sequence
= ( ) is weighted statistically convergent to , then it is strongly ( , )-summable to provided that | − | is bounded; that is, there exists a constant such that | − | ≤ for all ∈ N. We write for the set of all sequences = ( ) which are strongly ( , )-summable to . Then, for any ∈ C[ , ].

Rate of Weighted -Statistical Convergence
First we define the rate of weighted -statistically convergent sequence as follows.
Definition 10. Let = ( , ) be a nonnegative regular matrix and let ( ) be a positive nonincreasing sequence. Then, a sequence = ( ) is weighted -statistically convergent to with the rate of ( ) if for each > 0 where In symbol, we will write We will prove the following auxiliary result by using the above definition.
Given > 0, define It is easy to see that This yields that holds for all ∈ N. Since = max{ , }, (59) gives that Taking limit → ∞ in (60) together with (56), we obtain Thus, Similarly, we can prove (ii) and (iii).
Proof. Equation (27) can be reformed into the following form: where = ‖ ‖ ∞ . For a given > 0, we will define the following sets: It follows from (67) that holds for ∈ N. Since = max{ , }, we obtain from (69) that , .

Example and the Concluding Remark
The